# Definition of matrix equation

Put content here**Definition: Matrix Equation**
⏎
A matrix equation is an equation of the form:
⏎
**`Ax = b`**
⏎
where:
- `A` is an `m × n` matrix (the coefficient matrix)
- `x` is an `n × 1` column vector (the unknown)
- `b` is an `m × 1` column vector (the constant)
⏎
**Interpretation:** The product `Ax` is defined as the linear combination of the columns of `A` using the entries of `x` as weights:
```
Ax = x₁·col₁(A) + x₂·col₂(A) + ... + xₙ·colₙ(A)
```
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**Example:** If `A = [[1, 2], [3, 4]]` and `x = [5, 6]ᵀ`, then:
```
Ax = 5·[1, 3]ᵀ + 6·[2, 4]ᵀ = [5, 15]ᵀ + [12, 24]ᵀ = [17, 39]ᵀ
```
⏎
This definition bridges the algebraic form of linear systems with the geometric concept of linear combinations and span.

# Parents

* Matrix equations